Torus fibrations, gerbes, and duality
Author(s)
Bibliographic Information
Torus fibrations, gerbes, and duality
（Memoirs of the American Mathematical Society, no. 901）
American Mathematical Society, 2008
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Note
Includes bibliographical references (p. 87-90)
Description and Table of Contents
Description
Let $X$ be a smooth elliptic fibration over a smooth base $B$. Under mild assumptions, the authors establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an $\mathcal{O}{\times}$ gerbe over a genus one fibration which is a twisted form of $X$. The roles of the gerbe and the twist are interchanged by the authors' duality. The authors state a general conjecture extending this to allow singular fibers, and they prove the conjecture when $X$ is a surface. The duality extends to an action of the full modular group. This duality is related to the Strominger-Yau-Zaslow version of mirror symmetry, to twisted sheaves, and to non-commutative geometry.
Table of Contents
Introduction The Brauer group and the Tate-Shafarevich group Smooth genus one fibrations Surfaces Modified $T$-duality and the SYZ conjecture Appendix A. Duality for representations of $1$-motives, by Dmitry Arinkin Bibliography.
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